Asymptotic Expansion of the Solutions to a Regularized Boussinesq System (Theory and Numerics) - Safa, Ahmad; Le Meur, Hervé; Chehab, Jean-Paul; Talhouk, Raafat (Acta applicandae mathematicae, 2024)
We consider the propagation of surface water waves described by the Boussinesq system. Following (Molinet et al. in Nonlinearity 34:744–775, 2021 ), we introduce a regularized Boussinesq system obtained by adding a non-local pseudo-differential operator define by g λ [ ζ ] ˆ = | k | λ ζ ˆ k with λ ∈ ] 0 , 2 ] . In this paper, we display a twofold approach: first, we study theoretically the existence of an asymptotic expansion for the solution to the Cauchy problem associated to this regularized Boussinesq system with respect to the regularizing parameter ϵ . Then, we compute numerically the function coefficients of the expansion (in ϵ ) and verify numerically the validity of this expansion up to order 2. We also check the numerical L 2 stability of the numerical algorithm.